**1. Introduction**

The onboard antennas of the returned spacecraft are subjected to intensive aerodynamic heating when the spacecraft passes through the dense layers of the atmosphere [1]. In these conditions, radio-transparent heat-resistant thermal protection is used to protect the antennas from external influences. The open ends of the transmission lines are used as the emitter to obtain a wide directional pattern. The most offer used radiation from the open end of the round waveguide. In the first approximation, we consider a flat uniform thermal protection. Under the conditions of aerodynamic heating, the electrical parameters of thermal protection significantly change (relative permittivity ε and tangent of the dielectric loss angle tgδ). These changes lead to a noticeable increase in absorption losses in the heat shield, reflection from its boundaries, as well as to the appearance of surface and

side waves. Together, all this leads to a change in the directional pattern and a decrease in the efficiency of the onboard antenna.

Evaluation of these changes is absolutely necessary to determine the radio technical characteristics of the onboard radio equipment. The paper solves the problem of determining the characteristics of the radiation of a circular waveguide through a heat shield subjected to aerodynamic heating.

that is, in fact, the slow-moving processes of heating of the dielectric thermal protection are considered. This approach and methods of solution were used in a

*Characteristics of Radiation of a round Waveguide through a Flat Homogeneous Heat Shield*

The problem can be formulated as a boundary with respect to the tangent magnetic field in the material, and with respect to the tangent electric field. The second method is more convenient because of the simple form of boundary condi-

The magnetic component of the electromagnetic field *Hy* at z ≥ 0 must satisfy

*∂*2 *Hy <sup>∂</sup>z*<sup>2</sup> <sup>þ</sup> *<sup>k</sup>*<sup>2</sup>

We apply the Fourier transform for x and y coordinates to equation (Eq. (1)).

*<sup>x</sup>* � *<sup>k</sup>*<sup>2</sup> *y*

*Hy*ð Þ *<sup>x</sup>*, *<sup>y</sup>*, 0 exp �*j kxx* <sup>þ</sup> *kyy* � � � � *dxdy:*

Solution of equation (Eq. (2)) satisfying the radiation conditions (for z ≥*d*) has the form for area 1, that is, the area occupied by the dielectric plate (0 < z < d),

> *z* � � <sup>þ</sup> *<sup>L</sup>* exp *jkz*<sup>1</sup>

*<sup>y</sup>* <sup>¼</sup> *<sup>M</sup>* exp �*jkzz* � �,

Reasoning in a similar way, with respect to the tangent magnetic component of

*z* � � <sup>þ</sup> *<sup>B</sup>* exp *jkz*<sup>1</sup>

*<sup>x</sup>* <sup>¼</sup> *<sup>C</sup>*exp �*jkzz* � �*:*

Satisfying the equations arising from the Maxwell equation, we obtain expres-

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>k</sup>*<sup>2</sup> � *<sup>k</sup>*<sup>2</sup>

*<sup>x</sup>* � *<sup>k</sup>*<sup>2</sup> *y*

*:*

� �*H*^ *<sup>y</sup>* <sup>¼</sup> 0, (2)

*z* � �*:*

*z* � �,

where *ε*= *ε*<sup>1</sup> for 0 ≤ z ≤ d, *ε* = 1 for z > d, and k is the wave number.

*<sup>ε</sup>* � *<sup>k</sup>*<sup>2</sup>

*εHy* ¼ 0, (1)

the following wave equation in a Cartesian coordinate system x, y, z:

*∂*2 *Hy ∂y*<sup>2</sup> þ

*∂*2 *Hy ∂x*<sup>2</sup> þ

*∂*2 *H*^ *y <sup>∂</sup>z*<sup>2</sup> <sup>þ</sup> *<sup>k</sup>*<sup>2</sup>

ð ð<sup>∞</sup>

�∞

*<sup>H</sup>*^ ð Þ<sup>1</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*<sup>H</sup>*^ ð Þ<sup>1</sup>

sions for the spectral components of the electric field

*<sup>x</sup>* � *<sup>k</sup>*<sup>2</sup> *y*

*k*2 <sup>ε</sup> � *<sup>k</sup>*<sup>2</sup>

q

*<sup>y</sup>* ¼ *D* exp �*jkz*<sup>1</sup>

*<sup>H</sup>*^ ð Þ<sup>2</sup>

; *kz* ¼

*<sup>x</sup>* ¼ *A* exp �*jkz*<sup>1</sup>

*<sup>H</sup>*^ ð Þ<sup>2</sup>

the field, we obtain the following equations for the spectral component:

q

For region 2, that is, the region behind the plate (z ≥ d), we get

where is the direct Fourier transform,

*<sup>H</sup>*^ *<sup>y</sup>* <sup>¼</sup>

number of works, for example [7].

*DOI: http://dx.doi.org/10.5772/intechopen.92036*

**2. Main part**

tions for z = 0 [8].

where *kz*<sup>1</sup> ¼

**169**

We get

The problems of calculating the interaction of the onboard antenna with a heatshielding dielectric insert are very difficult and poorly developed. The development of mathematical models of antenna windows is reduced to solving an external problem of electrodynamics—electromagnetic excitation of bodies or diffraction of radio waves. At the same time, we will use well-known analytical methods of solution. The radio technical characteristics of the antenna window, for which we obtain an analytical description, include a radiation pattern, radiation conductivity, antenna temperature, and a number of other characteristics that describe more subtle electrodynamic effects, as well as energy characteristics.

In theoretical terms, the electrodynamic problem in general can be formulated as follows. There is a radiating, open antenna a, located on an infinite screen, in front of which is a dielectric layer of thickness d with a complex permittivity-bridge εα(x, y, z, t) (see **Figure 1**).

In this general formulation, the solution of the electrodynamic problem is associated with significant mathematical difficulties, the main problem being the need to solve the Maxwell equations for an arbitrary law of change in the parameters of media in space and time. With some simplifying assumptions, the problem was solved in the ray approximation. In [2], the wave front method is used to analyze the radiation diagram of an antenna covered by a dielectric layer. In [3–5], the method proposed in [6] is used to find the radiation diagram, according to which the antenna radiation diagram in the presence of an infinite flat dielectric layer is simply multiplied by the diagram in the free space by the flat wave transmission coefficient for the flat layer, taking into account the corresponding angle of arrival and the plane of polarization of the wave.

At their core, all these methods are close to each other and are essentially based on the approximation of geometric optics, which is true in the quasi-optical domain. In relation to the problem under consideration (the resonance region), a strict solution of the Maxwell equations is required. From analytical methods of the solution, it is possible to apply the method of integral transformations and the method of eigenfunctions. Both of these methods will be used in the future. In this case, we assume that the parameters of thermal protection do not depend on time,

*Characteristics of Radiation of a round Waveguide through a Flat Homogeneous Heat Shield DOI: http://dx.doi.org/10.5772/intechopen.92036*

that is, in fact, the slow-moving processes of heating of the dielectric thermal protection are considered. This approach and methods of solution were used in a number of works, for example [7].
